[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/jp\/wiki10\/archives\/322000#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/jp\/wiki10\/archives\/322000","headline":"\u7403\u9762\u4e09\u89d2\u6cd5 – Wikipedia","name":"\u7403\u9762\u4e09\u89d2\u6cd5 – Wikipedia","description":"before-content-x4 \u7403\u9762\u4e09\u89d2\u5f62 after-content-x4 \u7403\u9762\u4e09\u89d2\u6cd5\uff08\u304d\u3085\u3046\u3081\u3093\u3055\u3093\u304b\u304f\u307b\u3046\u3001\u82f1: spherical trigonometry\uff09\u3068\u306f\u3001\u3044\u304f\u3064\u304b\u306e\u5927\u5186\u3067\u56f2\u307e\u308c\u305f\u7403\u9762\u4e0a\u306e\u56f3\u5f62\uff08\u7403\u9762\u591a\u89d2\u5f62\u3001\u3068\u304f\u306b\u7403\u9762\u4e09\u89d2\u5f62\uff09\u306e\u8fba\u3084\u89d2\u306e\u4e09\u89d2\u95a2\u6570\u9593\u306e\u95a2\u4fc2\u3092\u6271\u3046\u7403\u9762\u5e7e\u4f55\u5b66\u306e\u4e00\u5206\u91ce\u3067\u3042\u308b\u3002 \u7403\u9762\u4e0a\u306b2\u70b9A,B\u304c\u3042\u308b\u3068\u304d\u3001\u3053\u306e2\u70b9\u3068\u7403\u306e\u4e2d\u5fc3\u3092\u901a\u308b\u5e73\u9762\u3067\u5207\u65ad\u3057\u305f\u3068\u304d\u306e\u65ad\u9762\u306b\u73fe\u308c\u308b\u5186\u304c\u5927\u5186\u3067\u3042\u308a\u3001\u3053\u306e\u3068\u304d\u306e\u5927\u5186\u4e0a\u306e\u5f27AB\u3092\u7403\u9762\u591a\u89d2\u5f62\u306b\u304a\u3044\u3066\u306f\u8fba\u3068\u547c\u3076\u3002 \u901a\u5e38\u3001\u7403\u306e\u534a\u5f84\u306f1\u3068\u3059\u308b\u306e\u3067\u3001\u8fba\u306e\u9577\u3055\u306f\u305d\u306e\u8fba\u3092\u542b\u3080\u5927\u5186\u306b\u304a\u3051\u308b\u4e2d\u5fc3\u89d2\u306e\u5f27\u5ea6\u6cd5\u8868\u793a\u3068\u4e00\u81f4\u3059\u308b\u3002 \u5e73\u9762\u4e09\u89d2\u6cd5\u3067\u306f6\u3064\u306e\u8981\u7d20\u306e\u3046\u30613\u3064\u306e\u8981\u7d20\u304c\u6c7a\u5b9a\u3055\u308c\u308c\u3070\u3001\u6b8b\u308a\u306e3\u3064\u306e\u8981\u7d20\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u7403\u9762\u4e09\u89d2\u6cd5\u3067\u3082\u540c\u69d8\u306b\u30013\u3064\u306e\u8981\u7d20\u304c\u5206\u304b\u308c\u3070\u6b8b\u308a\u306e3\u3064\u306e\u8981\u7d20\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b[1]\u3002 \u7403\u9762\u4e09\u89d2\u6cd5\u306f\u3001\u4e3b\u306b\u5929\u6587\u5b66\u3084\u822a\u6d77\u8853\u3067\u5229\u7528\u3055\u308c\u3066\u304d\u305f\u3002\u73fe\u5728\u3067\u306f\u96fb\u5b50\u8a08\u7b97\u6a5f\u306e\u767a\u9054\u306b\u3088\u308a\u3001\u3088\u308a\u7c21\u6f54\u306b\u5f0f\u3092\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u884c\u5217\u3092\u4f7f\u7528\u3057\u305f\u5ea7\u6a19\u5909\u63db\u306b\u8a08\u7b97\u65b9\u6cd5\u304c\u79fb\u884c\u3057\u3066\u3044\u308b[2]\u3002 after-content-x4 Table of Contents \u7403\u9762\u4e09\u89d2\u6cd5\u306e\u57fa\u672c\u516c\u5f0f[\u7de8\u96c6]\u8a98\u5c0e\u5b9a\u7406[\u7de8\u96c6]\u76f4\u89d2\u7403\u9762\u4e09\u89d2\u5f62[\u7de8\u96c6]\u30cd\u30a4\u30d4\u30a2\u306e\u5186[\u7de8\u96c6]\u8c61\u9650\u4e09\u89d2\u5f62[\u7de8\u96c6]\u6975\u4e09\u89d2\u5f62\u3068\u53cc\u5bfe\u539f\u7406[\u7de8\u96c6]haversine \u534a\u6b63\u77e2\u95a2\u6570[\u7de8\u96c6]\u30c9\u30e9\u30f3\u30d6\u30eb (Delambre) \u306e\u516c\u5f0f[\u7de8\u96c6]\u30cd\u30a4\u30d4\u30a2 (Napier) \u306e\u516c\u5f0f[\u7de8\u96c6]\u95a2\u9023\u9805\u76ee[\u7de8\u96c6]\u53c2\u8003\u6587\u732e[\u7de8\u96c6]","datePublished":"2022-04-18","dateModified":"2022-04-18","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/jp\/wiki10\/archives\/author\/lordneo#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/jp\/wiki10\/archives\/author\/lordneo","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/5\/50\/RechtwKugeldreieck.svg\/356px-RechtwKugeldreieck.svg.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/5\/50\/RechtwKugeldreieck.svg\/356px-RechtwKugeldreieck.svg.png","height":"358","width":"356"},"url":"https:\/\/wiki.edu.vn\/jp\/wiki10\/archives\/322000","about":["Wiki"],"wordCount":11999,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4  \u7403\u9762\u4e09\u89d2\u5f62       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4\u7403\u9762\u4e09\u89d2\u6cd5\uff08\u304d\u3085\u3046\u3081\u3093\u3055\u3093\u304b\u304f\u307b\u3046\u3001\u82f1: spherical trigonometry\uff09\u3068\u306f\u3001\u3044\u304f\u3064\u304b\u306e\u5927\u5186\u3067\u56f2\u307e\u308c\u305f\u7403\u9762\u4e0a\u306e\u56f3\u5f62\uff08\u7403\u9762\u591a\u89d2\u5f62\u3001\u3068\u304f\u306b\u7403\u9762\u4e09\u89d2\u5f62\uff09\u306e\u8fba\u3084\u89d2\u306e\u4e09\u89d2\u95a2\u6570\u9593\u306e\u95a2\u4fc2\u3092\u6271\u3046\u7403\u9762\u5e7e\u4f55\u5b66\u306e\u4e00\u5206\u91ce\u3067\u3042\u308b\u3002\u7403\u9762\u4e0a\u306b2\u70b9A,B\u304c\u3042\u308b\u3068\u304d\u3001\u3053\u306e2\u70b9\u3068\u7403\u306e\u4e2d\u5fc3\u3092\u901a\u308b\u5e73\u9762\u3067\u5207\u65ad\u3057\u305f\u3068\u304d\u306e\u65ad\u9762\u306b\u73fe\u308c\u308b\u5186\u304c\u5927\u5186\u3067\u3042\u308a\u3001\u3053\u306e\u3068\u304d\u306e\u5927\u5186\u4e0a\u306e\u5f27AB\u3092\u7403\u9762\u591a\u89d2\u5f62\u306b\u304a\u3044\u3066\u306f\u8fba\u3068\u547c\u3076\u3002\u901a\u5e38\u3001\u7403\u306e\u534a\u5f84\u306f1\u3068\u3059\u308b\u306e\u3067\u3001\u8fba\u306e\u9577\u3055\u306f\u305d\u306e\u8fba\u3092\u542b\u3080\u5927\u5186\u306b\u304a\u3051\u308b\u4e2d\u5fc3\u89d2\u306e\u5f27\u5ea6\u6cd5\u8868\u793a\u3068\u4e00\u81f4\u3059\u308b\u3002\u5e73\u9762\u4e09\u89d2\u6cd5\u3067\u306f6\u3064\u306e\u8981\u7d20\u306e\u3046\u30613\u3064\u306e\u8981\u7d20\u304c\u6c7a\u5b9a\u3055\u308c\u308c\u3070\u3001\u6b8b\u308a\u306e3\u3064\u306e\u8981\u7d20\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u7403\u9762\u4e09\u89d2\u6cd5\u3067\u3082\u540c\u69d8\u306b\u30013\u3064\u306e\u8981\u7d20\u304c\u5206\u304b\u308c\u3070\u6b8b\u308a\u306e3\u3064\u306e\u8981\u7d20\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b[1]\u3002\u7403\u9762\u4e09\u89d2\u6cd5\u306f\u3001\u4e3b\u306b\u5929\u6587\u5b66\u3084\u822a\u6d77\u8853\u3067\u5229\u7528\u3055\u308c\u3066\u304d\u305f\u3002\u73fe\u5728\u3067\u306f\u96fb\u5b50\u8a08\u7b97\u6a5f\u306e\u767a\u9054\u306b\u3088\u308a\u3001\u3088\u308a\u7c21\u6f54\u306b\u5f0f\u3092\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u884c\u5217\u3092\u4f7f\u7528\u3057\u305f\u5ea7\u6a19\u5909\u63db\u306b\u8a08\u7b97\u65b9\u6cd5\u304c\u79fb\u884c\u3057\u3066\u3044\u308b[2]\u3002       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Table of Contents\u7403\u9762\u4e09\u89d2\u6cd5\u306e\u57fa\u672c\u516c\u5f0f[\u7de8\u96c6]\u8a98\u5c0e\u5b9a\u7406[\u7de8\u96c6]\u76f4\u89d2\u7403\u9762\u4e09\u89d2\u5f62[\u7de8\u96c6]\u30cd\u30a4\u30d4\u30a2\u306e\u5186[\u7de8\u96c6]\u8c61\u9650\u4e09\u89d2\u5f62[\u7de8\u96c6]\u6975\u4e09\u89d2\u5f62\u3068\u53cc\u5bfe\u539f\u7406[\u7de8\u96c6]haversine \u534a\u6b63\u77e2\u95a2\u6570[\u7de8\u96c6]\u30c9\u30e9\u30f3\u30d6\u30eb (Delambre) \u306e\u516c\u5f0f[\u7de8\u96c6]\u30cd\u30a4\u30d4\u30a2 (Napier) \u306e\u516c\u5f0f[\u7de8\u96c6]\u95a2\u9023\u9805\u76ee[\u7de8\u96c6]\u53c2\u8003\u6587\u732e[\u7de8\u96c6]\u7403\u9762\u4e09\u89d2\u6cd5\u306e\u57fa\u672c\u516c\u5f0f[\u7de8\u96c6]ABC \u3092\u7403\u9762\u4e09\u89d2\u5f62\u3068\u3057\u8fba BC, CA, AB \u306e\u9577\u3055\u3092\u305d\u308c\u305e\u308c a, b, c \u3068\u3059\u308b\u3002\u5f27 AB \u3092\u542b\u3080\u5927\u5186\u304c\u4e57\u308b\u5e73\u9762\u3068\u5f27 AC \u3092\u542b\u3080\u5927\u5186\u304c\u4e57\u308b\u5e73\u9762\u306e\u306a\u3059\u89d2\u3092 A \u3068\u3059\u308b\u3002\u3053\u308c\u306f\u3001\u70b9 A \u306b\u304a\u3051\u308b2\u3064\u306e\u5927\u5186\u306e\u63a5\u30d9\u30af\u30c8\u30eb\u306e\u306a\u3059\u89d2\u3068\u3082\u3044\u3048\u308b\u3002\u305f\u3060\u3057\u3001a \u3068\u4e00\u81f4\u3059\u308b\u3068\u306f\u9650\u3089\u306a\u3044\u3002\u540c\u69d8\u306b B, C \u3082\u5b9a\u7fa9\u3059\u308b\u3002\u3053\u306e\u3068\u304d\u3001\u6b21\u306e\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3002\u7403\u9762\u4e09\u89d2\u6cd5\u306e\u4f59\u5f26\u5b9a\u7406       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4cos\u2061a=cos\u2061bcos\u2061c+sin\u2061bsin\u2061ccos\u2061Acos\u2061b=cos\u2061ccos\u2061a+sin\u2061csin\u2061acos\u2061Bcos\u2061c=cos\u2061acos\u2061b+sin\u2061asin\u2061bcos\u2061C{displaystyle {begin{aligned}cos a&=cos bcos c+sin bsin ccos A\\cos b&=cos ccos a+sin csin acos B\\cos c&=cos acos b+sin asin bcos Cend{aligned}}}\u7403\u9762\u4e09\u89d2\u6cd5\u306e\u6b63\u5f26\u5b9a\u7406sin\u2061asin\u2061A=sin\u2061bsin\u2061B=sin\u2061csin\u2061C{displaystyle {frac {sin a}{sin A}}={frac {sin b}{sin B}}={frac {sin c}{sin C}}}\u6b63\u5f26\u4f59\u5f26\u5b9a\u7406sin\u2061acos\u2061B=cos\u2061bsin\u2061c\u2212sin\u2061bcos\u2061ccos\u2061Asin\u2061acos\u2061C=cos\u2061csin\u2061b\u2212sin\u2061ccos\u2061bcos\u2061A{displaystyle {begin{aligned}sin acos B&=cos bsin c-sin bcos ccos A\\sin acos C&=cos csin b-sin ccos bcos Aend{aligned}}}\u7403\u9762\u4e09\u89d2\u6cd5\u306e\u6b63\u63a5\u5b9a\u7406tan\u2061A+B2tan\u2061A\u2212B2=tan\u2061a+b2tan\u2061a\u2212b2{displaystyle {frac {tan {dfrac {A+B}{2}}}{tan {dfrac {A-B}{2}}}}={frac {tan {dfrac {a+b}{2}}}{tan {dfrac {a-b}{2}}}}}\u7403\u9762\u4e09\u89d2\u6cd5\u306e\u4f59\u63a5\u5b9a\u7406cot\u2061asin\u2061b=cos\u2061bcos\u2061C+cot\u2061Asin\u2061C{displaystyle cot asin b=cos bcos C+cot Asin C}\u9762\u7a4d\uff08\u7403\u9762\u306e\u534a\u5f84 =r{displaystyle =r}\uff0c\u7403\u904e\u91cf (Spherical Excess) =E{displaystyle =E}\uff0c2s=a+b+c{displaystyle 2s=a+b+c}\uff09\u7403\u9762\u4e09\u89d2\u5f62ABC\u306e\u9762\u7a4d =Er2{displaystyle =Er^{2}}E=A+B+C\u2212\u03c0=4tan\u22121\u2061tan\u2061s2tan\u2061s\u2212a2tan\u2061s\u2212b2tan\u2061s\u2212c2=2sin\u22121\u2061sin\u2061ssin\u2061(s\u2212a)sin\u2061(s\u2212b)sin\u2061(s\u2212c)2cos\u2061a2cos\u2061b2cos\u2061c2=2cos\u22121\u20611+cos\u2061a+cos\u2061b+cos\u2061c4cos\u2061a2cos\u2061b2cos\u2061c2{displaystyle {begin{aligned}E&=A+B+C-pi \\&=4tan ^{-1}{sqrt {tan {frac {s}{2}}tan {frac {s-a}{2}}tan {frac {s-b}{2}}tan {frac {s-c}{2}}}}\\&=2sin ^{-1}{frac {sqrt {sin ssin(s-a)sin(s-b)sin(s-c)}}{2cos {dfrac {a}{2}}cos {dfrac {b}{2}}cos {dfrac {c}{2}}}}\\&=2cos ^{-1}{frac {1+cos a+cos b+cos c}{4cos {dfrac {a}{2}}cos {dfrac {b}{2}}cos {dfrac {c}{2}}}}end{aligned}}}\u7b2c1\u5f0f\u3092\u30b8\u30e9\u30fc\u30eb\uff08\u30d5\u30e9\u30f3\u30b9\u8a9e\u7248\u3001\u82f1\u8a9e\u7248\uff09\u306e\u5f0f\u3001\u7b2c2\u5f0f\u3092\u30ea\u30e5\u30a4\u30ea\u30a8\u306e\u5f0f\u3001\u7b2c3\u5f0f\u3092\u30ab\u30cb\u30e7\u30ea\uff08\u30a4\u30bf\u30ea\u30a2\u8a9e\u7248\u3001\u82f1\u8a9e\u7248\uff09\u306e\u5f0f\u3001\u7b2c4\u5f0f\u3092\u30aa\u30a4\u30e9\u30fc\u306e\u5f0f\u3068\u3044\u3046\u3002\u8a98\u5c0e\u5b9a\u7406[\u7de8\u96c6]2s=a+b+c{displaystyle 2s=a+b+c} \u3001 2S=A+B+C{displaystyle 2S=A+B+C} \u3068\u304a\u304f\u3002sin\u2061A2=sin\u2061(s\u2212b)sin\u2061(s\u2212c)sin\u2061bsin\u2061ccos\u2061A2=sin\u2061ssin\u2061(s\u2212a)sin\u2061bsin\u2061ctan\u2061A2=sin\u2061(s\u2212b)sin\u2061(s\u2212c)sin\u2061ssin\u2061(s\u2212a){displaystyle {begin{aligned}sin {frac {A}{2}}&={sqrt {frac {sin(s-b)sin(s-c)}{sin bsin c}}}\\cos {frac {A}{2}}&={sqrt {frac {sin ssin(s-a)}{sin bsin c}}}\\tan {frac {A}{2}}&={sqrt {frac {sin(s-b)sin(s-c)}{sin ssin(s-a)}}}end{aligned}}}sin\u2061a2=\u2212cos\u2061Scos\u2061(S\u2212A)sin\u2061Bsin\u2061Ccos\u2061a2=cos\u2061(S\u2212B)sin\u2061(S\u2212C)sin\u2061Bsin\u2061Ctan\u2061a2=\u2212cos\u2061Scos\u2061(S\u2212A)cos\u2061(S\u2212B)cos\u2061(S\u2212C){displaystyle {begin{aligned}sin {frac {a}{2}}&={sqrt {frac {-cos Scos(S-A)}{sin Bsin C}}}\\cos {frac {a}{2}}&={sqrt {frac {cos(S-B)sin(S-C)}{sin Bsin C}}}\\tan {frac {a}{2}}&={sqrt {frac {-cos Scos(S-A)}{cos(S-B)cos(S-C)}}}end{aligned}}}cos\u2061acos\u2061C=sin\u2061acot\u2061b\u2212sin\u2061Ccot\u2061B{displaystyle cos acos C=sin acot b-sin Ccot B}\u76f4\u89d2\u7403\u9762\u4e09\u89d2\u5f62[\u7de8\u96c6]\u5929\u6587\u5b66\u3084\u822a\u6d77\u8853\u3067\u306f\u4e00\u3064\u306e\u89d2\u304c\u76f4\u89d2\u306e\u5834\u5408\u304c\u591a\u304f\u3001\u3053\u306e\u5834\u5408\u516c\u5f0f\u306f\u7c21\u5358\u306b\u306a\u308b[3]\u3002C=\u03c02{displaystyle C={frac {pi }{2}}} \u3068\u3059\u308b\u3068\u3001(R1)cos\u2061c=cos\u2061acos\u2061b,(R6)tan\u2061b=cos\u2061Atan\u2061c,(R2)sin\u2061a=sin\u2061Asin\u2061c,(R7)tan\u2061a=cos\u2061Btan\u2061c,(R3)sin\u2061b=sin\u2061Bsin\u2061c,(R8)cos\u2061A=sin\u2061Bcos\u2061a,(R4)tan\u2061a=tan\u2061Asin\u2061b,(R9)cos\u2061B=sin\u2061Acos\u2061b,(R5)tan\u2061b=tan\u2061Bsin\u2061a,(R10)cos\u2061c=cot\u2061Acot\u2061B.{displaystyle {begin{alignedat}{4}&{text{(R1)}}&qquad cos c&=cos a,cos b,&qquad qquad &{text{(R6)}}&qquad tan b&=cos A,tan c,\\&{text{(R2)}}&sin a&=sin A,sin c,&&{text{(R7)}}&tan a&=cos B,tan c,\\&{text{(R3)}}&sin b&=sin B,sin c,&&{text{(R8)}}&cos A&=sin B,cos a,\\&{text{(R4)}}&tan a&=tan A,sin b,&&{text{(R9)}}&cos B&=sin A,cos b,\\&{text{(R5)}}&tan b&=tan B,sin a,&&{text{(R10)}}&cos c&=cot A,cot B.end{alignedat}}}\u3053\u308c\u3089\u3092\u8a18\u61b6\u3059\u308b\u305f\u3081\u306b\u30cd\u30a4\u30d4\u30a2\u306e\u5186\u304c\u3042\u308b\u3002\u30cd\u30a4\u30d4\u30a2\u306e\u5186[\u7de8\u96c6]  \u30cd\u30a4\u30d4\u30a2\u306e\u5186\u3068\u76f4\u89d2\u7403\u9762\u4e09\u89d2\u5f62\u53f3\u56f3\u3092\u30cd\u30a4\u30d4\u30a2\u306e\u5186\u3068\u3044\u3046\u3002 a\u00af=\u03c02\u2212a,b\u00af=\u03c02\u2212b{displaystyle {bar {a}}={frac {pi }{2}}-a,{bar {b}}={frac {pi }{2}}-b}\u3067\u3042\u308b\u3002\u30cd\u30a4\u30d4\u30a2\u306e\u5186\u306e\u3069\u308c\u304b\u4e00\u3064\u306e\u8981\u7d20\u3092\u4e2d\u592e\u8981\u7d20\u3068\u3057\u3001\u305d\u306e\u96a3\u306e\u8981\u7d20\u3092\u96a3\u63a5\u8981\u7d20\u3001\u6b8b\u308a\u306e\u4e2d\u592e\u8981\u7d20\u306e\u53cd\u5bfe\u5074\u306b\u3042\u308b2\u3064\u306e\u8981\u7d20\u3092\u5bfe\u5411\u8981\u7d20\u3068\u3059\u308b\u3002\u3053\u306e\u3068\u304d\u4e0a\u8a18\u306e\u5b9a\u7406(R1)\uff5e(R10)\u306f\u6b21\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3002\u4e2d\u592e\u8981\u7d20\u306e\u4f59\u5f26 = \u96a3\u63a5\u8981\u7d20\u306e\u4f59\u63a5\u306e\u7a4d\u4e2d\u592e\u8981\u7d20\u306e\u4f59\u5f26 = \u5bfe\u5411\u8981\u7d20\u306e\u6b63\u5f26\u306e\u7a4d\u8c61\u9650\u4e09\u89d2\u5f62[\u7de8\u96c6]\u7403\u9762\u4e09\u89d2\u5f62\u306e\u4e00\u8fba\u304c\u03c02{displaystyle {frac {pi }{2}}}\u3068\u306a\u3063\u3066\u3044\u308b\u3082\u306e\u3092\u8c61\u9650\u4e09\u89d2\u5f62\u3068\u3044\u3046\u3002\u3053\u306e\u5834\u5408\u3082\u516c\u5f0f\u306f\u7c21\u5358\u306b\u306a\u308b[4]\u3002c=\u03c02{displaystyle c={frac {pi }{2}}}\u3068\u3059\u308b\u3068\u3001(Q1)cos\u2061C=\u2212cos\u2061Acos\u2061B,(Q6)tan\u2061B=\u2212cos\u2061atan\u2061C,(Q2)sin\u2061A=sin\u2061asin\u2061C,(Q7)tan\u2061A=\u2212cos\u2061btan\u2061C,(Q3)sin\u2061B=sin\u2061bsin\u2061C,(Q8)cos\u2061a=sin\u2061bcos\u2061A,(Q4)tan\u2061A=tan\u2061asin\u2061B,(Q9)cos\u2061b=sin\u2061acos\u2061B,(Q5)tan\u2061B=tan\u2061bsin\u2061A,(Q10)cos\u2061C=\u2212cot\u2061acot\u2061b.{displaystyle {begin{alignedat}{4}&{text{(Q1)}}&qquad cos C&=-cos A,cos B,&qquad qquad &{text{(Q6)}}&qquad tan B&=-cos a,tan C,\\&{text{(Q2)}}&sin A&=sin a,sin C,&&{text{(Q7)}}&tan A&=-cos b,tan C,\\&{text{(Q3)}}&sin B&=sin b,sin C,&&{text{(Q8)}}&cos a&=sin b,cos A,\\&{text{(Q4)}}&tan A&=tan a,sin B,&&{text{(Q9)}}&cos b&=sin a,cos B,\\&{text{(Q5)}}&tan B&=tan b,sin A,&&{text{(Q10)}}&cos C&=-cot a,cot b.end{alignedat}}}\u8c61\u9650\u4e09\u89d2\u5f62\u306e\u5834\u5408\u306f\u30cd\u30a4\u30d4\u30a2\u306e\u5186\u306b A\u00af,B\u00af,a,\u03c0\u2212C,b{displaystyle {bar {A}},{bar {B}},a,pi -C,b} \u3092\u3042\u3066\u306f\u3081\u308c\u3070\u3088\u3044\u3002\u6975\u4e09\u89d2\u5f62\u3068\u53cc\u5bfe\u539f\u7406[\u7de8\u96c6]  \u7403\u9762\u4e09\u89d2\u5f62 ABC \u306e\u6975\u4e09\u89d2\u5f62 A’B’C’ \u4e00\u822c\u306b\u3001\u5927\u5186\u306e\u5e73\u9762\u306b\u5782\u76f4\u306a\u76f4\u5f84\u306e\u4e21\u7aef\u3092\u305d\u306e\u5927\u5186\u306e\u6975\u3068\u3044\u3046\u3002\u53f3\u56f3\u306b\u304a\u3044\u3066\u7403\u9762\u4e09\u89d2\u5f62ABC\u306e1\u3064\u306e\u8fbaBC\u3092\u8003\u3048\u308b\u3068\u3001\u305d\u308c\u306b\u306f2\u3064\u306e\u6975\u304c\u3042\u308b\u304c\u3001\u305d\u306e\u3046\u3061\u8fbaBC\u304b\u3089\u898b\u3066A\u3068\u540c\u3058\u5074\u306b\u3042\u308b\u307b\u3046\u3092A’\u3068\u3059\u308b\u3002\u540c\u69d8\u306b\u8fbaCA, AB\u306b\u3064\u3044\u3066\u3082\u6975B’, C’\u3092\u5b9a\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u3053\u306e\u3088\u3046\u306b\u3057\u3066\u5f97\u3089\u308c\u305f3\u70b9A’, B’, C’\u3092\u7d50\u3093\u3067\u65b0\u3057\u3044\u4e00\u3064\u306e\u7403\u9762\u4e09\u89d2\u5f62A’B’C’\u304c\u5f97\u3089\u308c\u308b\u3002\u3053\u308c\u3092\u5143\u306e\u7403\u9762\u4e09\u89d2\u5f62ABC\u306e\u6975\u4e09\u89d2\u5f62\u3068\u3044\u3046\u3002\u7403\u9762\u4e09\u89d2\u5f62A’B’C’\u304c\u7403\u9762\u4e09\u89d2\u5f62ABC\u306e\u6975\u4e09\u89d2\u5f62\u3067\u3042\u308b\u306a\u3089\u3070\u3001\u9006\u306b\u7403\u9762\u4e09\u89d2\u5f62ABC\u306f\u7403\u9762\u4e09\u89d2\u5f62A’B’C’\u306e\u6975\u4e09\u89d2\u5f62\u3067\u3042\u308b\u3002\u307e\u305f\u4eca\u3001\u7403\u9762\u4e09\u89d2\u5f62A’B’C’\u304c\u7403\u9762\u4e09\u89d2\u5f62ABC\u306e\u6975\u4e09\u89d2\u5f62\u3067\u3042\u308b\u3068\u3057\u3001\u305d\u306e\u4e09\u8fba\u3001\u4e09\u89d2\u3092\u305d\u308c\u305e\u308ca’, b’, c’\u3001A’, B’, C’\u3067\u8868\u3059\u3068\u3001a, b, c\u3001A, B, C\u3068\u306e\u9593\u306b\u306f\u6b21\u306e\u3088\u3046\u306a\u95a2\u4fc2\u304c\u3042\u308b[5]\uff1aA\u2032=\u03c0\u2212a,B\u2032=\u03c0\u2212b,C\u2032=\u03c0\u2212c,a\u2032=\u03c0\u2212A,b\u2032=\u03c0\u2212B,c\u2032=\u03c0\u2212C.{displaystyle {begin{alignedat}{3}A’&=pi -a,&qquad B’&=pi -b,&qquad C’&=pi -c,\\a’&=pi -A,&b’&=pi -B,&c’&=pi -C.end{alignedat}}}\u4e0a\u8a18\u3092\u307e\u3068\u3081\u308b\u3068\u3001\u7403\u9762\u4e09\u89d2\u5f62\u306e\u6cd5\u5247\u306f\u3001\u305d\u308c\u305e\u308c\u306e\u8981\u7d20\u306e\u5411\u304b\u3044\u5408\u3063\u305f\u8981\u7d20\u306e\u88dc\u89d2\u306b\u7f6e\u304d\u63db\u3048\u3066\u3082\u6210\u308a\u7acb\u3064\u3002\u3053\u308c\u3092\u53cc\u5bfe\u539f\u7406\u3068\u3044\u3046[6]\u3002\u5177\u4f53\u4f8b\u3092\u3042\u3052\u308b\u3068cos\u2061a=cos\u2061bcos\u2061c+sin\u2061bsin\u2061ccos\u2061A{displaystyle cos a=cos bcos c+sin bsin ccos A}\u304b\u3089cos\u2061(\u03c0\u2212A)=cos\u2061(\u03c0\u2212B)cos\u2061(\u03c0\u2212C)+sin\u2061(\u03c0\u2212B)sin\u2061(\u03c0\u2212C)cos\u2061(\u03c0\u2212a){displaystyle cos(pi -A)=cos(pi -B)cos(pi -C)+sin(pi -B)sin(pi -C)cos(pi -a)}\u3059\u306a\u308f\u3061cos\u2061A=\u2212cos\u2061Bcos\u2061C+sin\u2061Bsin\u2061Ccos\u2061a{displaystyle cos A=-cos Bcos C+sin Bsin Ccos a}\u304c\u6210\u308a\u7acb\u3064\u3002haversine \u534a\u6b63\u77e2\u95a2\u6570[\u7de8\u96c6]hav\u2061x\u00a0=def\u00a012(1\u2212cos\u2061x)=sin2\u2061x2{displaystyle operatorname {hav} x {overset {underset {mathrm {def} }{}}{=}} {frac {1}{2}}(1-cos x)=sin ^{2}{frac {x}{2}}}\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u534a\u6b63\u77e2\u95a2\u6570 hav\u2061(){displaystyle operatorname {hav} ()} \u304c\u822a\u6d77\u7528\u3068\u3057\u3066\u4f7f\u7528\u3055\u308c\u3066\u3044\u305f\u3002\u5177\u4f53\u7684\u306b\u306f\u3001\u7403\u9762\u4e0a\u306e2\u70b9\u9593\u306e\u7403\u9762\u306b\u6cbf\u3063\u305f\u8ddd\u96e2\u3092\u6c42\u3081\u308b\u5834\u9762\u3067\u3042\u308b\u3002\u524d\u8ff0\u306e\u4f59\u5f26\u5b9a\u7406\u3067\u3082\u6c42\u3081\u308b\u3053\u3068\u306f\u53ef\u80fd\u3060\u304c\u30012\u5730\u70b9\u9593\u304c\u8fd1\u3044\uff08\u4f8b\u3048\u3070\u7d4c\u5ea6\u5dee\u304c0\u306b\u8fd1\u3044\uff09\u3068\u304dcos\u2061x=0.99999999{displaystyle cos {x}=0.99999999}\u3068\u3044\u3063\u305f\u5024\u3092\u4f7f\u3046\u3053\u3068\u306b\u306a\u308a\u3001\u8a08\u7b97\u3057\u3065\u3089\u3044\u306e\u3067\u3053\u3061\u3089\u3092\u7528\u3044\u305f\u3002\u5b9a\u7fa9\u304b\u3089\u3053\u306e\u95a2\u6570\u306e\u5024\u306f\u5e38\u306b\u6b63\u3067\u3042\u308a\u3001\u304b\u3064\u3001\u5076\u95a2\u6570\u3067\u3042\u308b\u3002cos\u2061x=1\u22122hav\u2061x{displaystyle cos x=1-2operatorname {hav} x} \u304b\u3089\u3001\u6700\u521d\u306e\u7403\u9762\u4e09\u89d2\u6cd5\u306e\u4f59\u5f26\u5b9a\u7406\u3092\u66f8\u304d\u76f4\u3059\u30681\u22122hav\u2061a=cos\u2061bcos\u2061c+sin\u2061bsin\u2061c(1\u22122hav\u2061A){displaystyle 1-2operatorname {hav} a=cos bcos c+sin bsin c(1-2operatorname {hav} A)}\u3088\u308ahav\u2061a\u00a0=hav\u2061(b\u2212c)+sin\u2061bsin\u2061c\u00a0hav\u2061A{displaystyle operatorname {hav} a =operatorname {hav} (b-c)+sin bsin c operatorname {hav} A}\u3068\u306a\u308b\u3002\u534a\u5f841\u306e\u7403\u306e2\u70b9\u306e\u7def\u5ea6\u304c \u03c61,\u03c62{displaystyle varphi _{1},varphi _{2}}\u3001\u7d4c\u5ea6\u304c \u03bb1,\u03bb2{displaystyle lambda _{1},lambda _{2}} \u3067\u3042\u308b\u3068\u304d\u30012\u70b9\u9593\u306e\u5927\u5186\u4e0a\u306e\u8ddd\u96e2\u3092 \u03b8{displaystyle theta } \u3068\u3059\u308b\u3068\u3001hav\u2061\u03b8=hav\u2061(\u03c62\u2212\u03c61)+cos\u2061\u03c62cos\u2061\u03c61hav\u2061(\u03bb2\u2212\u03bb1){displaystyle operatorname {hav} theta =operatorname {hav} (varphi _{2}-varphi _{1})+cos varphi _{2}cos varphi _{1}operatorname {hav} (lambda _{2}-lambda _{1})}\u3053\u3053\u304b\u3089\u6c42\u3081\u305f \u03b8{displaystyle theta } \u306b\u5730\u7403\u306e\u534a\u5f84\u7d046371km\u3092\u639b\u3051\u308c\u3070\u3001\u5730\u7403\u4e0a\u3067\u306e\u304a\u304a\u3088\u305d\u306e\u8ddd\u96e2\u304c\u5206\u304b\u308b\u3002\u30c9\u30e9\u30f3\u30d6\u30eb (Delambre) \u306e\u516c\u5f0f[\u7de8\u96c6]\u30b8\u30e3\u30f3\uff1d\u30d0\u30c6\u30a3\u30b9\u30c8\u30fb\u30b8\u30e7\u30bc\u30d5\u30fb\u30c9\u30e9\u30f3\u30d6\u30eb\u306b\u3088\u308b\u3002\u30ac\u30a6\u30b9\u306e\u516c\u5f0f\u3068\u547c\u3070\u308c\u308b\u3053\u3068\u3082\u3042\u308b\u3002cos\u2061A+B2cos\u2061c2=cos\u2061a+b2sin\u2061C2sin\u2061A+B2cos\u2061c2=cos\u2061a\u2212b2cos\u2061C2cos\u2061A\u2212B2sin\u2061c2=sin\u2061a+b2sin\u2061C2sin\u2061A\u2212B2sin\u2061c2=sin\u2061a\u2212b2cos\u2061C2{displaystyle {begin{aligned}cos {frac {A+B}{2}}cos {frac {c}{2}}&=cos {frac {a+b}{2}}sin {frac {C}{2}}\\sin {frac {A+B}{2}}cos {frac {c}{2}}&=cos {frac {a-b}{2}}cos {frac {C}{2}}\\cos {frac {A-B}{2}}sin {frac {c}{2}}&=sin {frac {a+b}{2}}sin {frac {C}{2}}\\sin {frac {A-B}{2}}sin {frac {c}{2}}&=sin {frac {a-b}{2}}cos {frac {C}{2}}end{aligned}}}\u30cd\u30a4\u30d4\u30a2 (Napier) \u306e\u516c\u5f0f[\u7de8\u96c6]tan\u2061A+B2=cos\u2061a\u2212b2cos\u2061a+b2cot\u2061C2tan\u2061A\u2212B2=sin\u2061a\u2212b2sin\u2061a+b2cot\u2061C2tan\u2061a+b2=cos\u2061A\u2212B2cos\u2061A+B2tan\u2061c2tan\u2061a\u2212b2=sin\u2061A\u2212B2sin\u2061A+B2tan\u2061c2{displaystyle {begin{aligned}tan {frac {A+B}{2}}={frac {cos {dfrac {a-b}{2}}}{cos {dfrac {a+b}{2}}}}cot {frac {C}{2}}\\tan {frac {A-B}{2}}={frac {sin {dfrac {a-b}{2}}}{sin {dfrac {a+b}{2}}}}cot {frac {C}{2}}\\tan {frac {a+b}{2}}={frac {cos {dfrac {A-B}{2}}}{cos {dfrac {A+B}{2}}}}tan {frac {c}{2}}\\tan {frac {a-b}{2}}={frac {sin {dfrac {A-B}{2}}}{sin {dfrac {A+B}{2}}}}tan {frac {c}{2}}end{aligned}}}\u95a2\u9023\u9805\u76ee[\u7de8\u96c6]^ \u6e21\u8fba\u654f\u592b \u300e\u6570\u7406\u5929\u6587\u5b66\u300f \u6052\u661f\u793e\u539a\u751f\u95a3 p.41^ \u300e\u5929\u4f53\u306e\u4f4d\u7f6e\u8a08\u7b97\u300f\u3001\u9577\u6ca2\u5de5\u3001\u5730\u4eba\u66f8\u9928 p.12-32^ \u6e21\u8fba\u654f\u592b \u300e\u6570\u7406\u5929\u6587\u5b66\u300f \u6052\u661f\u793e\u539a\u751f\u95a3 p.49^ \u6e21\u8fba\u654f\u592b \u300e\u6570\u7406\u5929\u6587\u5b66\u300f \u6052\u661f\u793e\u539a\u751f\u95a3 p.50^ AB\u306e\u5927\u5186\u4e0a\u306e\u5ef6\u9577\u3068B’C’\u3068\u306e\u4ea4\u70b9\u3092E\u3001CA\u306e\u5927\u5186\u4e0a\u306e\u5ef6\u9577\u3068B’C’\u3068\u306e\u4ea4\u70b9\u3092F\u3068\u3059\u308b\u3068\u3001(B\u2032E+EF)+(EF+FC\u2032)=B\u2032C\u2032+EF=a\u2032+A=\u03c0{displaystyle (B’E+EF)+(EF+FC’)=B’C’+EF=a’+A=pi } \u3067\u3042\u308b\u3053\u3068\u304c\u5bb9\u6613\u306b\u5206\u304b\u308b\u3002\u6b8b\u308a\u306e\u95a2\u4fc2\u3082\u540c\u69d8\u306b\u793a\u3055\u308c\u308b\u3002^ \u6e21\u8fba\u654f\u592b \u300e\u6570\u7406\u5929\u6587\u5b66\u300f \u6052\u661f\u793e\u539a\u751f\u95a3 p.52\u53c2\u8003\u6587\u732e[\u7de8\u96c6]     (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki10\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki10\/archives\/322000#breadcrumbitem","name":"\u7403\u9762\u4e09\u89d2\u6cd5 – Wikipedia"}}]}]